TITLE: Analytic representatives for Kahler classes
QUESTION [6 upvotes]: If we are given  compact complex manifold $X$ and a Kahler class $[\omega]$, 
can we always find a positive definite representative $\omega \in [\omega]$ that is 
real analytic?

REPLY [10 votes]: Yes.  Run the Kähler-Ricci heat flow 
$$
\frac{\mathrm{d}}{\mathrm{d}t}\left(\omega(t)\right) = -\mathrm{Ric}\bigl(\omega(t)\bigr)
$$ 
with initial condition $\omega(0) = \omega$. This will exist for some time interval $[0,T)$, and the $\omega_t$ for $t>0$ will all be be real-analytic with respect to the natural real-analytic structure on $M$ given by the complex structure.  Note that we have
$$
\left[\omega(t)\right] = \left[\omega(0)\right] - t\,c_1(M).
$$
Now, for $0<t_1<t_2<T$, with $t_1$ very small with respect to $t_2$, consider the $2$-form
$$
\omega(t_1,t_2) = \frac{\omega(t_1) - (t_1/t_2)\ \omega(t_2)}{1-t_1/t_2}
$$
It is easy to see that, if $t_1/t_2>0$ is very small, then $\omega(t_1,t_2)$ is a positive $(1,1)$-form.  It is real-analytic and, by construction, it satisfies
$$
\left[\omega(t_1,t_2)\right] = \left[\omega(0)\right] = [\omega].
$$